Fr\'echet means, conceptually appealing, generalize the Euclidean expectation to general metric spaces. We explore how well Fr\'echet means can be estimated from independent and identically distributed samples and uncover a fundamental limitation: In the vicinity of a probability distribution $P$ with nonunique means, independent of sample size, it is not possible to uniformly estimate Fr\'echet means below a precision determined by the diameter of the set of Fr\'echet means of $P$. Implications were previously identified for empirical plug-in estimators as part of the phenomenon \emph{finite sample smeariness}. Our findings thus confirm inevitable statistical challenges in the estimation of Fr\'echet means on metric spaces for which there exist distributions with nonunique means. Illustrating the relevance of our lower bound, examples of extrinsic, intrinsic, Procrustes, diffusion and Wasserstein means showcase either deteriorating constants or slow convergence rates of empirical Fr\'echet means for samples near the regime of nonunique means.

翻译：Fréchet均值在概念上具有吸引力，它将欧几里得期望推广到一般度量空间。我们探究了如何从独立同分布样本中估计Fréchet均值，并揭示了一个基本限制：在具有非唯一均值的概率分布$P$附近，无论样本量大小如何，都无法以低于由$P$的Fréchet均值集合直径所确定的精度来一致地估计Fréchet均值。这一结论先前已在经验插值估计中被识别为“有限样本平滑度”现象的一部分。因此，我们的发现证实了在度量空间上估计Fréchet均值时不可避免的统计挑战，因为在这些空间中存在具有非唯一均值的分布。为了说明我们下界的相关性，外在均值、内在均值、Procrustes均值、扩散均值和Wasserstein均值的实例展示了，在非唯一均值区域附近的样本中，经验Fréchet均值要么常数退化，要么收敛速度缓慢。